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NASA NTRS1999

S-Wave Dispersion Relations: Exact Left Hand E-Plane Discontinuity from the Born Series

We show, for a superposition of Yukawa potentials, that the left hand cut discontinuity in the complex E plane of the (S-wave) scattering amplitude is given exactly, in an interval depending on n, by the discontinuity of the Born series stopped at order n. This also establishes an inverse and unexpected correspondence of the Born series at positive high energies and negative low energies. We can thus construct a viable dispersion relation (DR) for the partial (S-) wave amplitude. The high numerical precision achievable by the DR is demonstrated for the exponential potential at zero scattering energy. We also briefly discuss the extension of our results to Field Theory.

Bessis, D.↗
NASA NTRS1989

A space-time discretization procedure for wave propagation problems

Higher order compact algorithms are developed for the numerical simulation of wave propagation by using the concept of a discrete dispersion relation. The dispersion relation is the imprint of any linear operator in space-time. The discrete dispersion relation is derived from the continuous dispersion relation by examining the process by which locally plane waves propagate through a chosen grid. The exponential structure of the discrete dispersion relation suggests an efficient splitting of convective and diffusive terms for dissipative waves. Fourth- and eighth-order convection schemes are examined that involve only three or five spatial grid points. These algorithms are subject to the same restrictions that govern the use of dispersion relations in the constructions of asymptotic expansions to nonlinear evolution equations. A new eighth-order scheme is developed that is exact for Courant numbers of 1, 2, 3, and 4. Examples are given of a pulse and step wave with a small amount of physical diffusion.

Davis, Sanford↗
NASA NTRS1995

Acceleration and heating of heavy ions in high speed solar wind streams

Left hand polarized Alfven waves generated in coronal holes propagate in the direction of high speed solar wind streams, accelerating and heating heavy ions. As the solar wind expands, the ratio between the frequency of the Alfven waves and the proton gyrofrequency increases, due to the decrease of the interplanetary magnetic field, and encounter first the local ion gyrofrequency of the species with the largest M(sub l) = m(sub l)/z(sub l)m(sub p) (m(sub l) is the mass of species l, m(sub p) is the proton mass and z(sub l) is the degree of ionization of species l). It is shown that the Alfven waves experience there strong absorption and cannot propagate any further until the ions are accelerated and heated. Once this occurs, the Alfven waves continue to propagate until they meet the gyrofrequency of the next species giving rise to a similar phenomenon. In order to show this contention, we use the linear dispersion relation of ion cyclotron waves in a multicomponent plasma consisting of oxygen ions, alpha particles and protons. We assume that at any distance from the sun, the Alfven waves follow the local dispersion relation of electromagnetic ion cyclotron waves. To illustrate the results, we solve the dispersion relation for oxygen ions and alpha particles drifting relative to the protons. The dispersion relation has three branches. The first branch starts at zero frequency and goes to the Doppler-shifted oxygen ion gyrofrequency. The second branch starts close to the oxygen gyrofrequency, and goes to the Doppler-shifted alpha particle gyrofrequency. The third branch starts close to the alpha particle gyrofrequency, and goes to the proton gyrofrequency. The Alfven waves propagate following the first branch of the dispersion relation. When they reach the Doppler-shifted oxygen ion gyrofrequency, the ions are accelerated and heated to some definite values. When these values are reached, the dispersion relation changes, and it is now the first branch of the dispersion relation, the one which goes to the Doppler-shifted alpha particle gyrofrequency. The Alfven waves continue to propagate along the first branch of the dispersion relation and proceed to accelerate and heat the alpha particles.

Gomberoff, L.↗
NASA NTRS1992

Comments on absolute and convective instabilities

The paper puts into question the tenability of a critique by Oscarsson and Roennmark (1986) regarding the theory of absolute and convective instabilities. The space-time evolution of instabilities is reviewed with attention given to the two categories of time-asymptotic evolution. It is mentioned that the notion of time-asymptotics does not simply refer to t approaching infinity and that the evolution of the arbitrary initial perturbation at t = 0 is obtained by convolving the Green's function. The examples used in the previous critique describe instabilities that evolve in time only and are concluded to be nongeneric for continuous media, and the corresponding treatment of a time-asymptotic limit is characterized by imprecision. The original authors respond by suggesting that nondispersive waves are physically legitimate as in common plasma-wave modes and Alfven waves and can be described by their dispersion relation. A dispersive term is added to the relation so that it can be applied to ion-acoustic waves and other phenomena.

Ram, A. K.↗
NASA NTRS1993

Beyond Clausius-Mossotti - Wave propagation on a polarizable point lattice and the discrete dipole approximation

We derive the dispersion relation for electromagnetic waves propagating on a lattice of polarizable points. From this dispersion relation we obtain a prescription for choosing dipole polarizabilities so that an infinite lattice with finite lattice spacing will mimic a continuum with dielectric constant. The discrete dipole approximation is used to calculate scattering and absorption by a finite target by replacing the target with an array of point dipoles. We compare different prescriptions for determining the dipole polarizabilities. We show that the most accurate results are obtained when the lattice dispersion relation is used to set the polarizabilities.

Draine, B. T.↗
NASA NTRS1985

Theory of the current-driven ion cyclotron instability in the bottomside ionosphere

A comprehensive treatment of the theory of electrostatic ion cyclotron instability in the collisional bottomside ionosphere is given. The linear dispersion relations for the ion cyclotron instability (electron and ion collisions) are derived. Analytical solutions to the dispersion relation are obtained in three limits: collisionless; weakly collisional; and collisional. In addition, the linear dispersion relation is solved numerically, using parameters typical of the high-latitude bottomside ionosphere. It is pointed out that the linear growth rates in the bottomside collisional regime can be of the same order as the corresponding topside collisionless growth rates. A nonlinear saturation mechanism for the electrostatic ion cyclotron instability is proposed on the basis of Dupree's (1966) resonance broadening theory.

Satyanarayana, P.↗
NASA NTRS1973

Ion streaming instabilities with application to collisionless shock wave structure

The electromagnetic dispersion relation for two counterstreaming ion beams of arbitrary relative strength flowing parallel to a dc magnetic field is derived. The beams flow through a stationary electron background and the dispersion relation in the fluid approximation is unaffected by the electron thermal pressure. The dispersion relation is solved with a zero net current condition applied and the regions of instability in the k-U space (U is the relative velocity between the two ion beams) are presented. The parameters are then chosen to be applicable for parallel shocks. It was found that unstable waves with zero group velocity in the shock frame can exist near the leading edge of the shock for upstream Alfven Mach numbers greater than 5.5. It is suggested that this mechanism could generate sufficient turbulence within the shock layer to scatter the incoming ions and create the required dissipation for intermediate strength shocks.

Golden, K. I.↗
NASA NTRS2016

Wave Telescope Technique for MMS Magnetometer

Multipoint measurements are a powerful method in studying wavefields in space plasmas.The wave telescope technique is tested against magnetic field fluctuations in the terrestrial magnetosheath measured by the four Magnetospheric Multiscale (MMS) spacecraft on a spatial scale of about 20 km.The dispersion relation diagram and the wave vector distribution are determined for the first time in the ion-kinetic range. Moreover, the dispersion relation diagram is determined in a proxy plasma restframe by regarding the low-frequency dispersion relation as a Doppler relation and compensating for the apparent phase velocity. Fluctuations are highly compressible, and the wave vectors have an angle of about 60 from the mean magnetic field. We interpret that the measured fluctuations represent akinetic-drift mirror mode in the magnetosheath which is dispersive and in a turbulent state accompanied by a sideband formation.

Magnetometer↗
NASA NTRS1973

Ion streaming instabilities with application to collisionless shock wave structure

The electromagnetic dispersion relation for two counterstreaming ion beams of arbitrary relative strength flowing parallel to a dc magnetic field is derived. The beams flow through a stationary electron background and the dispersion relation in the fluid approximation is unaffected by the electron thermal pressure. Magnetic effects on the ion beams are included, but the electrons are treated as a magnetized fluid. The dispersion relation is solved with a zero net current condition applied and the regions of instability in the k-U space (U is the relative velocity between the two ion beams) are presented. These results are extensions of Kovner's analysis for weak beams. The parameters are then chosen to be applicable for parallel shocks. It is found that unstable waves with zero group velocity in the shock frame can exist near the leading edge of the shock for upstream Alfven Mach numbers greater than 5.5.

Golden, K. I.↗
NASA NTRS1985

Energy dependence of cosmic ray composition above 10(15) GeV/nucleus

It is argued that above 10 to the 5th power GeV/nucleus, in the range where charge-resolved spectra have not yet been determined, the appropriate measures of equal-energy composition are 1nA and 1nA , the mean value and dispersion relative to the mean value and dispersion relative to the mean of 1nA, where A is the mass number. Experimental data which are sensitive to changes in 1nA with increasing energy are examined. It is found that, taken as a whole, they show no change (+ or 0.5) between 10 to the 5th power and 10 to the 6th power GeV, and a decrease of 1.5 + or - 0.5 between 10 to the 6th power and 10 to the 8th power GeV, with no further change + or - 0.5) above 10 to the 8th power GeV. Taken as a whole, the various indirect estimates of the absolute value of 1nA above 10 to the 5th power GeV/nucleus are also consistent with this pattern. For a wide range of astrophysically plausible composition models the value of the other measure, 1nA is insensitive to changes in 1nA . Because of this the existing data on 1nA can likewise easily be reconciled with this pattern.

Linsley, J.↗
NASA NTRS1996

[Kelvin-Helmholtz instability in protostellar jets]

NASA grant NAG 5 2866, funded by the Astrophysics Theory Program, enabled the study the Kelvin-Helmholtz instability in protostellar jets. In collaboration with co-investigator Philip Hardee, the PI derived the analytic dispersion relation for the instability in including a cooling term in the energy equation which was modeled as one of two different power laws. Numerical solutions to this dispersion relation over a wide range of perturbation frequencies, and for a variety of parameter values characterizing the jet (such as Mach number, and density ratio) were found It was found that the growth rates and wavelengths associated with unstable roots of the dispersion relation in cooling jets are significantly different than those associated with adiabatic jets, which have been studied previously. In collaboration with graduate student Jianjun Xu (funded as a research associate under this grant), hydrodynamical simulations were used to follow the growth of the instability into the nonlinear regime. It was found that asymmetric surface waves lead to large amplitude, sinusoidal distortions of the jet, and ultimately to disruption Asymmetric body waves, on the other hand, result in the formation of shocks in the jet beam in the nonlinear regime. In cooling jets, these shocks lead to the formation of dense knots and filaments of gas within the jet. For sufficiently high perturbation frequencies, however, the jet cannot respond and it remains symmetric. Applying these results to observed systems, such as the Herbig-Haro jets HH34, HH111 and HH47 which have been observed with the Hubble Space Telescope, we predicted that some of the asymmetric structures observed in these systems could be attributed to the K-H modes, but that perturbations on timescales associated with the inner disk (about 1 year) would be too rapid to cause disruption. Moreover, it was discovered that weak shock 'spurs' in the ambient gas produced by ripples in the jet surface due to nonlinear, modes of surface and/or body waves could accelerate the ambient gas to low velocity. This latter effect represents a new mechanism by which supersonic jets can accelerate low velocity outflows.

Stone, James↗
NASA NTRS1997

The Stability of Radiatively Cooling Jets I. Linear Analysis

The results of a spatial stability analysis of a two-dimensional slab jet, in which optically thin radiative cooling is dynamically important, are presented. We study both magnetized and unmagnetized jets at external Mach numbers of 5 and 20. We model the cooling rate by using two different cooling curves: one appropriate to interstellar gas, and the other to photoionized gas of reduced metallicity. Thus, our results will be applicable to both protostellar (Herbig-Haro) jets and optical jets from active galactic nuclei. We present analytical solutions to the dispersion relations in useful limits and solve the dispersion relations numerically over a broad range of perturbation frequencies. We find that the growth rates and wavelengths of the unstable Kelvin-Helmholtz (K-H) modes are significantly different from the adiabatic limit, and that the form of the cooling function strongly affects the results. In particular, if the cooling curve is a steep function of temperature in the neighborhood of the equilibrium state, then the growth of K-H modes is reduced relative to the adiabatic jet. On the other hand, if the cooling curve is a shallow function of temperature, then the growth of K-H modes can be enhanced relative to the adiabatic jet by the increase in cooling relative to heating in overdense regions. Inclusion of a dynamically important magnetic field does not strongly modify the important differences between an adiabatic jet and a cooling jet, provided the jet is highly supermagnetosonic and not magnetic pressure-dominated. In the latter case, the unstable modes behave more like the transmagnetosonic magnetic pressure-dominated adiabatic limit. We also plot fluid displacement surfaces associated with the various waves in a cooling jet in order to predict the structures that might arise in the nonlinear regime. This analysis predicts that low-frequency surface waves and the lowest order body modes will be the most effective at producing observable features in the jet.

Hardee, Philip E.↗
NASA NTRS1983

Doubly diffusive magnetic buoyancy instability in the solar interior

An investigation of the buoyancy of diffuse magnetic fields has shown that in the presence of rotation, static equilibrium configurations of the toroidal magnetic field and ambient plasma can exist. In that case, the escape of toroidal magnetic flux from the solar interior may be determined by the growth of instabilities which the equilibrium configuration may be subject to. In connection with the present investigation, it is assumed that in the region of toroidal magnetic flux amplification, the magnetic field has not as yet filamented into flux ropes, and is therefore 'diffuse'. A study is conducted of the MHD stability of an electrically conducting and differentially rotating gas in the presence of a toroidal magnetic field, an external constant gravitational field, and radiance pressure. The full dispersion relation for the magnetic buoyancy problem is developed, and the solutions of the dispersion relation are discussed.

Schmitt, J. H. M. M.↗